I want to take a network of roads and lanes and reduce them to a schematic that shows their relationships but not their geographic shape.

A similar concept has been used for the London Underground map where the distances between stations are not too scale, it simply shows their relationships by vertices.

In my ideal world the roads would snap to a horizontal/vertical grid and the distances between them normalised. I am a Postgis/QuantumGIS user and want to know which tools or capabilities I should use to (for example) create a grid from Openstreetmap way data?

The method presented is used for area subdivisions, but the "rectilinearization" algorithm could be reused for network data. It should be more simple since for your case there is no constrain on the preservation of the areas - only the topology has to be preserved.